Null spaces and ranges for the classical and spherical Radon transforms

Let R be the classical Radon transform that integrates a function over hyperplanes in Rⁿ and let SM be the transform that integrates a function over spheres containing the origin in Rⁿ. We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in Rⁿ as well as the null space of SM for square integrable functions on a ball. We show SM: L²(Rⁿ) → L²(Rⁿ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g(x) is a Schwartz function on Rⁿ that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM(C^∞(Rⁿ)). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.     

Optimal broadcast on parallel locality models

In this paper matching upper and lower bounds for broadcast on general purpose parallel computation models that exploit network locality are proven. These models try to capture both the general purpose properties of models like the PRAM or BSP on the one hand, and to exploit network locality of special purpose models like meshes, hypercubes, etc., on the other hand. They do so by charging a cost l(|i−j|) for a communication between processors i and j, where l is a suitably chosen latency function.An upper bound T(p)=∑_i=0^loglogp2ⁱ·l(p^1/2i) on the runtime of a broadcast on a p processor H-PRAM is given, for an arbitrary latency function l(k).The main contribution of the paper is a matching lower bound, holding for all latency functions in the range from l(k)=Ω(logk/loglogk) to l(k)=O(log²k). This is not a severe restriction since for latency functions l(k)=O(logk/log¹⁺log(k)) with arbitrary >0, the runtime of the algorithm matches the trivial lower bound Ω(logp) and for l(k)=Θ(log¹⁺k) or l(k)=Θ(k), the runtime matches the other trivial lower bound Ω(l(p)). Both upper and lower bounds apply for other parallel locality models like Y-PRAM, D-BSP and E-BSP, too.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Exposed points and extremal problems in H1 on a bidisc

An essential bounded function ? gives a continuous linear functional on the Hardy space H¹ on the bitorus. In this paper, we consider extremal problems on H¹ when ? is a rational function, ? is a product of one variable functions or ? = |f|/f for some outer function f in H¹ such that f(z, w) has a good property with respect to w for a.e. z. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s‐Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 ‐ o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n^{1+Ω(1/log log n)}. Recall that the Ising model with inverse temperature β defined on a graph G = (V,E) is the distribution over {±}^Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < ∞ and the Ising model defined on G (n, d/n), there exists a β_d > 0, such that for all β < β_d with probability going to 1 as n →∞, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd?s‐Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub‐graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and β < β_d, where d tanh(β_d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Automorphisms for Skew PBW Extensions and Skew Quantum Polynomial Rings

In this work, I study the automorphisms of skew PBW extensions and skew quantum polynomials. I use Artamonov's works as reference for getting the principal results about automorphisms for generic skew PBW extensions and generic skew quantum polynomials. In general, if I have an endomorphism on a generic skew PBW extension and there are some x _i , x _j , x _u such that the endomorphism is not zero on these elements and the principal coefficients are invertible, then endomorphisms act over x _i as a _i x _i for some a _i in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with r = 0, as such Artamonov shows in his work. We use this result for giving some more general results for skew PBW extensions, using filtred-graded techniques. Finally, I use localization to characterize some class the endomorphisms and automorphisms for skew PBW extensions and skew quantum polynomials over Ore domains.     

Solvability of differential equations on open subsets of the Sierpiński gasket

We give necessary and sufficient conditions on the polynomial p for the differential equation p(Δ)u = f, based on the Laplacian, to be solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p, we find the open subsets on which p(Δ)u = f is solvable for any continuous f.     

A partially penalized P1/CR immersed finite element method for planar elasticity interface problems

We propose a partially penalized P₁/CR immersed finite element (IFE) method with midpoint values on edges as degrees of freedom for CR elements to solve planar elasticity interface problems. Optimal approximation errors in L² norm and H¹ semi‐norm are obtained for the P₁/CR IFE spaces. Moreover, by adding some stabilization terms on the edges of interface elements, we derive an optimal error estimate for the P₁/CR IFE method. Our method differs from the method with average values on edges as degrees of freedom for P₁/CR elements in Qin et al.'s study, where no approximation theoretical result was presented. Numerical examples confirm our theoretical results.     

A spectrum result on maximal partial ovoids of the generalized quadrangle , even

This article presents a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q even. We prove that for every integer k in an interval of, roughly, size [q²/10,9q²/10], there exists a maximal partial ovoid of size k on Q(4,q), q even. Since the generalized quadrangle W(q), q even, defined by a symplectic polarity of PG(3,q) is isomorphic to the generalized quadrangle Q(4,q), q even, the same result is obtained for maximal partial ovoids of W(q), q even. As equivalent results, the same spectrum result is obtained for minimal blocking sets with respect to planes of PG(3,q), q even, and for maximal partial 1-systems of lines on the Klein quadric Q⁺(5,q), q even.     

Diffraction Problems and the Extension of Hs–Functions

In this paper we derive a formula for the extension for a function from the Sobolev space H_s( R ⁿ₊) on the whole space H_s( R ⁿ), using the principle of diffraction on the hyperplane x_n = 0 for any real s.     

Symmetric invariant cocycles on the duals of q-deformations

We prove that for q∈C^? not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of Uqg is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of Uqg and Ug is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence.     

Pre Stieltjes Functions

In this paper we characterize the class C_k{{\mathcal{C}_k}} of functions f on (0,∞) for which f(x), . . . ,(x ^k f(x))^(k) are completely monotonic for given k. In the limit we obtain the well-known characterization of the class of Stieltjes functions as those functions f defined on the positive half line for which (x ^k f(x))^(k) is completely monotonic on (0,∞) for all k ≥ 0.     

p -Sparse BEM for weakly singular integral equation with random data

We consider the weakly singular boundary integral equation 𝒱u = g on a randomly perturbed smooth closed surface Γ(ω) with deterministic g or on a deterministic closed surface Γ with stochastic g (ω). The aim is the computation of the centered moments ℳ︁^k u, k ⩾ 1, if the corresponding moments of the perturbation are known. The problem on the stochastic surface is reduced to a problem on the nominal deterministic surface Γ with the random perturbation parameter κ (ω). Resulting formulation for the k th moment is posed in the tensor product Sobolev spaces and involve the k -fold tensor product operators. The standard full tensor product Galerkin BEM requires 𝒪(N^k) unknowns for the k th moment problem, where N is the number of unknowns needed to discretize the nominal surface Γ. Based on [1], we develop the p -sparse grid Galerkin BEM to reduce the number of unknowns to 𝒪(N (log N)^{k –1}) (cf. [2], [3] for the wavelet approach). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

The Swendsen‐Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is conjectured to converge to equilibrium in O(|V|^1/4) steps at any (inverse) temperature β, yet there are few results providing o(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, when β < β_c(d) where β_c(d) denotes the uniqueness/nonuniqueness threshold on infinite d‐regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a monotone version of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW dynamics has mixing time and relaxation time Θ(1) on any graph of maximum degree d for all β < β_c(d). Our proof technology can be applied to general monotone Markov chains, including for example the heat‐bath block dynamics, for which we obtain new tight mixing time bounds.     

In response to the article by Ivan Kausz: A discriminant and an upper bound for w 2 for hyperelliptic arithmetic surfaces

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.     

Special domains and nonmanipulability

We investigate domains on which a nonmanipulable, nondictatorial social choice function exists, having at least three distinct values. We do not make the assumptions of Kalai and Muller (1977). We classify all such 2-person functions on the domain which is the cyclic group Z_m. We show that for any domain containing Z_m, existence for 2 voters and existence for some n > 2 voters are equivalent. We show that for an n-person, onto, nonmanipulable social choice function F on Z_m, F(P₁, P₂,…, P_n) {x₁, x₂,…, x_n} always, x_i being the most preferred alternative under preference P_i. We show that no domain containing the dihedral group admits such a social choice function. We show that there exists a domain on which all k-tuples are free for arbitrarily large k, for which such a social choice function does exist.     

Monitoring process variability using auxiliary information

In this study a Shewhart type control chart namely V _r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V _r chart is developed and its comparison is made with the well-known Shewhart control chart namely S ² chart used for the same purpose. Using power curves as a performance measure it is observed that V _r chart outperforms the S ² chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ _yx . These efficiency conditions on ρ _yx are also obtained for V _r chart in this study.